What is a continued fraction?

A continued fraction expresses a number as an integer plus the reciprocal of another integer plus the reciprocal of another, and so on. It is written compactly as [a0; a1, a2, ...] and gives the most natural sequence of rational approximations to a real number.

What are convergents?

Convergents are the fractions you get by truncating the continued fraction after each term. Each convergent is the best rational approximation of the number for its denominator size, and they alternate above and below the true value while getting steadily closer.

How is each term computed?

Take the integer (floor) part as the first term, subtract it, take the reciprocal of the leftover fractional part, and repeat. The integer parts produced at each step are the continued fraction terms.

Why does the expansion of pi start with 3, 7, 15, 1?

Those are the floor values at each step. Truncating after 3 and 7 gives 22/7, and after 3, 7, 15, 1 gives 355/113, both classic approximations of pi. The large term 15 is why 22/7 is already quite accurate.

Why might the expansion stop early?

If the number is rational its expansion is finite and terminates exactly. For decimals entered with limited precision the remainder eventually becomes negligible, so the tool stops to avoid reporting noise from floating-point rounding.

Continued Fraction Converter

A continued fraction writes a number as an integer plus a reciprocal of an integer plus a reciprocal, nesting indefinitely. It produces the cleanest possible ladder of rational approximations — the convergents — which is why it underlies calendar design, gear ratios, and the best rational approximations of constants like pi.

How it works

The expansion is built by a simple repeat: take the floor, subtract it, reciprocate the remaining fractional part, and go again.

pi = 3.14159265...
  a0 = 3   remainder 0.14159...  ->  1/0.14159 = 7.0625...
  a1 = 7   remainder 0.0625...   ->  1/0.0625  = 15.99...
  a2 = 15  ...
  pi = [3; 7, 15, 1, ...]

Each truncation [a0; a1, ..., ak] is a convergent p_k / q_k, computed with the recurrence p_k = a_k * p_(k-1) + p_(k-2) and q_k = a_k * q_(k-1) + q_(k-2). The convergents are the best rational approximations for their denominator size and alternate just above and just below the true value, tightening at every step.

Tips and notes

The output uses the standard bracket notation with a semicolon after the integer part, for example [3; 7, 15, 1]. Each listed convergent shows its decimal value and absolute error so you can pick the trade-off you want between simplicity and accuracy. Because the input is read as a floating-point decimal, very long expansions eventually reflect rounding rather than the true number, so keep the term count modest unless you entered a high-precision value.